The cube below has sides of length 4 feet. If a cylindrical section of radius 2 feet is removed from the solid, what is the total remaining volume of the cube? Express your answer in cubic feet in terms of $\pi$.

[asy]
import solids; size(150); import three; defaultpen(linewidth(0.8)); currentprojection = orthographic(4,2.5,3);

draw((1,-1,0)--(1,1,0)--(-1,1,0)); draw((-1,1,0)--(-1,-1,0)--(1,-1,0), dashed);
draw((1,-1,2)--(1,1,2)--(-1,1,2)--(-1,-1,2)--cycle);
draw((1,-1,0)--(1,-1,2)); draw((1,1,0)--(1,1,2)); draw((-1,-1,0)--(-1,-1,2),dashed); draw((-1,1,0)--(-1,1,2));revolution c = cylinder((0,0,0), 1, 2);
draw(c,black);
[/asy]
The cube has volume $4^3=64$ cubic feet.  The cylinder has radius 2, height 4, and volume $\pi(2^2)(4)=16\pi$ cubic feet.  It follows that when the cylindrical section is removed from the solid, the remaining volume is $\boxed{64-16\pi}$ cubic feet.